*Today's guest writer is MindMover intern Lanz Lagman*

*, a student of BS Astronomy Technology from Rizal Technological University and a member of RTU Astronomy Society and Philippine Astronomical Society.*

Five years ago, Juno was launched from Cape Canaveral and its journey towards the largest planet of our Solar System began. A month ago in July 5, 2016, it successfully entered the planet's orbit as a part of its JOI (Jupiter Insertion Orbit) maneuver. Juno will slow down so that it will orbit Jupiter in two 53.5-day polar orbit periods (an orbit wherein the smaller body will pass by the poles of its parent body), before finally settling down to its sets of science orbits, that will provide us with high-quality data never seen before.

This mission aims to give humanity more insights about the evolution and secrets of Jupiter, using high-end instruments to peer within its vast clouds - just as the Roman goddess Juno spied on Jupiter and uncovered his mischievous acts.

**How did Juno reach Jupiter as NASA intended it?**

Several months after its launch five years ago, it performed Deep Space Maneuvers on August 30 and September 14, 2012 in order to refine its trajectory for an Earth flyby planned to occur on October 9, 2013. Juno proceeded towards Jupiter, and arrived on July 4, 2016.

Juno's trajectory. (Credit: NASA/JPL)

But wait, why does Juno need to loop around the Sun and pass near Earth on October 9, when it could have been launched on that date instead of August 2011? What's the purpose of making Juno loop around the Sun? Couldn't NASA just launch Juno at the date of the Earth flyby?

It was necessary for NASA's plan to increase Juno's velocity. The Deep Space Maneuvers and the Earth flyby were integral parts of a move called a gravitational slingshot, also known as gravity assist. Juno passed by Earth on October 9, with its closest approach within 559 kilometers from Earth's surface. Juno's instruments were tested during this encounter, and its velocity relative to the Sun increased significantly, just enough for Juno to arrive at Jupiter as scheduled.

As space agencies are commonly known to be bereft of funding, NASA has to carefully spend a measly $1.13 billion project investment to efficiently fulfill the goal of the Juno mission. Additionally, rocket fuel is very expensive; in order to carry more of it, rockets have to be much larger, and much heavier. That would not be economically efficient, considering that spacecraft traveling to outer planets are decelerated by the Sun.

If NASA insisted on spending more on fuel and strapping more engines, the billion-dollar fund would end up like Bing Bong from Inside Out. Unless our probes learn to do something similar to the Instant Transmission technique, we currently have no fuel-efficient engines powerful enough to send space probes as heavy as Juno directly to planets.

For comparison, New Horizons, the fastest spacecraft launched during its time, weighs only 478 kg, while Juno weighs 3625 kg. In the meantime, we're stuck with looping heavy spacecraft around planets for a speed boost. By doing this maneuver, Juno received a boost almost as powerful as a second rocket launch, for free!

For comparison, New Horizons, the fastest spacecraft launched during its time, weighs only 478 kg, while Juno weighs 3625 kg. In the meantime, we're stuck with looping heavy spacecraft around planets for a speed boost. By doing this maneuver, Juno received a boost almost as powerful as a second rocket launch, for free!

Juno's Earth flyby trajectory viewed from Earth's perspective, perpendicular to Earth's orbital plane.

(Credit: NASA's Eyes to the Solar System)

**How do gravitational slingshots work?**

Spacecraft seeking to increase their speeds have to steal a bit of momentum from a planet. Conservation of momentum and kinetic energy might seem to be violated, but since the planet is so massive, it's hardly affected, while the spacecraft gains a significant speed boost.

Imagine a fast-moving vehicle carrying billions of pesos dropping a thousand peso bill at your feet. You can pick it up and buy whatever you want with it; the vehicle may have lost money, but it's negligible compared to what it retains.

Imagine a fast-moving vehicle carrying billions of pesos dropping a thousand peso bill at your feet. You can pick it up and buy whatever you want with it; the vehicle may have lost money, but it's negligible compared to what it retains.

In this article, we will use vectors as our main tool to understand them. Vectors are defined as things that have magnitude and direction; their positions however, do not matter. We add vectors by joining the arrowhead of one vector to the tail of another, and we subtract them by flipping either of the two vectors to their opposite direction.

This means either two arrowheads are connected, or two tails. Afterwards, we must look at the flyby from two reference frames, one from the Earth's perspective, and the other from the Sun. The movement of a spacecraft aided by a gravity assist is best visualized by this animation.

This means either two arrowheads are connected, or two tails. Afterwards, we must look at the flyby from two reference frames, one from the Earth's perspective, and the other from the Sun. The movement of a spacecraft aided by a gravity assist is best visualized by this animation.

A sample spacecraft's trajectory as viewed from a planet and the Sun's reference frame.

(Credit: David Shortt)

First, let's differentiate the reference frame of the Earth from the Sun. At Earth's reference frame, the Earth is at the center, and hence, it stays at the same position when viewed from the top. The initial and final velocity (velocity is only a magnitude) of Juno with respect to it are therefore the same. At the Sun's reference frame, however, the Earth is moving. As a result, Juno's initial and final velocity at this frame would be different.

Vector components of Juno's initial conditions

In this diagram, Earth moves to the left, and its vector is defined as v ⃗_(

Juno's initial velocity with respect to Earth, going θ degrees from the vertical, is defined by the vector v ⃗_(

*E,S*) . We also define a reference line, called the vertical. It is shown as a broken line here, and it is similar to a y-axis. The direction of our vectors will be defined with respect to it except Earth's vector. We could assume that Earth is moving in a linear matter at both perspectives.Juno's initial velocity with respect to Earth, going θ degrees from the vertical, is defined by the vector v ⃗_(

*J,E,i*). In this case, v ⃗_(*J,S,i*), Juno's initial velocity with respect to the Sun's reference frame, going α degrees from the vertical, is the sum of the two previously mentioned vectors. To know whether Juno will gain or lose speed, let's look at the next diagram.
Vector components of Juno's final conditions

Earth's velocity vector does not change as previously explained, but when you look at Juno's final velocity vector relative to the Sun v ⃗_(

*J,S,f*), it increased. Similar to the previous situation, vectors are added when the head of one vector is connected to the tail of the other. As long as a spacecraft goes in the same direction as the planet, it will gain speed. If it goes toward the opposite side of the planet's velocity vector, it will slow down.
Mission planners who plan to insert their space probes into a planet's orbit will use gravity assist to increase their speed when the planet is farther than Earth from the sun, and slow down when that planet is nearer than Earth. Well-known missions that used gravity assists to increase their speed include Voyagers 1 and 2.

Voyager 2 used all planets from Mars to Neptune as catapults, Voyager 1 did not visit Uranus and Neptune. On the other hand, Mariner 2 and MESSENGER gave a part of their momentum to Earth to slow down, and insert themselves into the orbits of Mercury.

Voyager 2 used all planets from Mars to Neptune as catapults, Voyager 1 did not visit Uranus and Neptune. On the other hand, Mariner 2 and MESSENGER gave a part of their momentum to Earth to slow down, and insert themselves into the orbits of Mercury.

For the more advanced reader, here's the detailed calculation. Be warned however, equations ahead!

Before we start our calculations, we must define how accurate our calculations would be. To make our computations easier, we simplify the scenario to two dimensions instead of three. We use NASA's eyes to the Solar System to produce this image, and the view would be at the Earth's reference frame, perpendicular to its orbital plane. NASA has stated that the initial and final velocities of Juno with respect to the Sun are 78,000 mph and 93,000 mph respectively.

Earth's mean orbital velocity is 29.78 km/s or, as we convert and round off, 67,000 mph. Using an on-screen protractor, we could therefore measure θ and φ, which when rounded off to the nearest ones, yields 20° and 60° respectively. Lastly, the calculated values for velocities would be rounded off to the nearest thousands.

Since θ is 20° and the angle adjacent to its right is definitely 90°, its complement to the left side is 70°. The angle between v ⃗_(

*E,S*) and v ⃗_(*J,E,i*) is the supplement of 70°, which is 110°. The magnitude of a vector is indicated by a bar at each side. By using the sine law, we could therefore solve for α:
Equation (2) then yields 74° for α. Now we proceed to solve for |v ⃗_(

*J,E,i*)| , but first we must deconstruct the vector components of |v ⃗_(*J,S,i*)| and introduce equation (3):
Since the vertical is parallel to |v ⃗_(

*J,S,i*) | cos〖αy ̂ 〗, the angle between it and v ⃗_(*J,S,i*) would also be α. Now we also deconstruct v ⃗_(*J,E,i*) the same way as v ⃗_(*J,S,i*) and it also yields
By looking at the next illustration and with the help of equation (4), we could express |v ⃗_(

*J,E,i*)| in terms of |v ⃗_(*J,S,i*)| and α:
Since we already have the value for |v ⃗_(

*J,S,i*)| and α, we could now solve for |v ⃗_(*J,E,i*)| and we get 29,000 mph as an answer. We could proceed to solve for β and |v ⃗_(*J,S,f*)|. In order to do that, we must remember how the magnitudes of Juno's initial and final velocity with respect to Earth are related:Equation (6) is just the mathematical expression that their magnitudes are the same. We couldn't say that, however, to their respective directions. By looking at the next diagram, we could see how we can relate |v ⃗_(

*J,E,f*)| to v ⃗_(

*J,S,f*) and produce equation (7):

While this situation is very similar to the previous one, our approach will be different here compared to the previous initial diagrams. We could immediately solve for |v ⃗_(

*J,S,f*)| using cosine law:

At equation (8), 180° - (90° - φ ), which is equal to 150° refers to the angle between v ⃗_(

*J,E,f*) and v ⃗_(

*E,S*), which is the supplement of the complement of φ.

We now get 93,000 mph, exactly the estimate given by NASA.

From equation (9), we finally get β= 81°. Remember that it is just by coincidence that we arrived at the same value for |v ⃗_(

*J,S,f*)| due to the simplifications we earlier mentioned. How we rounded off our obtained values is very loose compared to how calculations in astrodynamics are done, wherein values in the thousandths are considered very important.REFERENCES:

1. Juno Earth Flyby. (2013, October 9). Retrieved from: http://www.nasa.gov/mission_pages/juno/earthflyby.html

2. Shortt, D. (2013, September 27). Gravity assist. Retrieved from http://www.planetary.org/blogs/guest-blogs/2013/20130926-gravity-assist.html

3. A Gravity Assist Primer. (n.d.) Retrieved from http://solarsystem.nasa.gov/basics/grav/primer.php

4. Cain, F. (2015, December 23). How Do Gravitational Slingshots Work? - Universe Today. Retrieved from: http://www.universetoday.com/113488/how-do-gravitational-slingshots-work/

5. Quick Facts. (n.d.). Retrieved from http://www.jpl.nasa.gov/news/press_kits/juno/facts

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